Trifurcated lined ducts: A comprehensive study on noise reduction strategies

The present research is centered on analyzing and modeling the scattering characteristics of a trifurcated waveguide that includes impedance discontinuities. A mode-matching method, grounded in projecting the solution onto orthogonal basis functions, is devised for the investigation. The impedance disparities at the interfaces are represented in normal velocity modes, which, when combined with pressure modes, result in a linear algebraic system. This system is subsequently truncated and inverted for numerical experimentation. The convergence of scattering amplitudes is assured by reconstructing matching conditions and adhering to conservation laws. The computational results indicate that optimizing attenuation behavior is achievable through manipulating variation bounding properties and impedance discontinuities.


Introduction
The theory of noise reduction has become a dynamic area of research due to large-scale industrial advancements.This research is crucial for various applications, including, aircraft and vehicle engines, turbofan engines, ducts, and pipes.Guided wave systems, known for their efficiency in carrying acoustic energy by preventing lateral diffusion, resist the decay of sound waves according to the inverse square law.
Numerous scientists and engineers have addressed noise reduction by considering different material properties of ducts and diverse geometrical designs.Rawlins [1] discussed noise reduction through a duct with a thin acoustical absorbent lining on parallel plates partitioning.According to Demir and Buyukaksoy [2], fixing the walls of a conduit with an acoustically lining material can fundamentally improve its acoustic performance.Morse [3] investigated the attenuation of sound in boundless shut pipes using acoustically absorbing liners.Subsequent analyses confirmed that fixing the properties of on the walls of waveguide enhances sound absorption.
The study of waveguides based on different mathematical formulations has been extensively discussed.Koch [4] introduced the Wiener-Hopf solution to specify the problem of the radiation of sound from a semi-infinite 2D channel with walls fixed with a responding sound retention substance.Jones [5] evaluated the far field and near-zone solutions for the issue of wave recasting of the differential system into a linear algebraic system that can be solved through inversion.
Precisely, the underlying problem provides a step further in generalizing the study of planar trifurcated lined ducts.The following sections make up the article: The basic waveguide structure is defined in section 2. In section 3, the mode-matching method is used to estimate the scattered field potentials in each region.The energy flux distribution in various regions is obtained in part 4 by numerically solving truncated infinite linear systems.In Section 5, numerical results are presented graphically.Section 6 summarizes the investigations.

Formulation of boundary value problem
The study focuses on the propagation of acoustic waves in a waveguide with partitions and impedance discontinuities.In a rectangular coordinate system (� x; � y), the waveguide can be divided into four regions defined as follows: • Region R 1 : � x < � 0; � y < j� aj Note that the bars in the variables represent the dimensional setting of those variables.The regions mentioned are filled with a compressible fluid with density ρ and sound speed c.In particular, region R 1 is bounded by rigid walls with an infinite impedance � Z 1 , while the surfaces of regions R 2 , R 3 , and R 4 have finite impedances � Z 2 , � Z 3 , and � Z 4 , respectively.Assuming a harmonic time dependence of e À io� t , where ω = ck represents the radian frequency with k being the wave number, the surface impedance � Z j can be expressed in terms of the time-independent fluid potential � φj, as mentioned in reference [31], that is where j = 1, 2, 3, 4 is used to specify the regions R j .The waveguide's time-independent fluid potential � φ is governed by the Helmholtz equation [31], which can be expressed as follows: The governing boundary value problem is non-dimensionalized using the length scale k −1 and time scale ω −1 such that x ¼ k� x; y ¼ k� y and illustrating the schematic configuration.The non-dimensional problem incorporates Helmholtz's equation with a unit wave number, which can be expressed as follows: The dimensionless form of boundary conditions are • At y = b, the specific impedance in dimensionless form

Solution methodology: Mode-matching approach
In order to understand the scattering properties of the given structure, we utilize the modematching technique to solve the governing boundary value problem.This technique involves obtaining the eigenfunction expansions of the duct regions and applying the matching interface conditions to convert the differential systems into linear algebraic systems.These linear algebraic systems are then truncated and inverted.In the subsequent subsections, we will explore the specifics of the eigenfunction expansions and the properties of the eigenfunctions in more detail.
0.1 In region R 1 : fx < 0; jyj < ag The acoustic region denoted as R 1 is enclosed by acoustically rigid boundaries described by Eq (4).Within this region, the propagation of sound satisfies the Helmholtz's equation as stated in Eq (3).To solve Eq (3) under the boundary condition (4), we employ the separation of variable method.This method allows us to decompose the solution into a series of eigenfunctions.The resulting solution takes the form of an eigenfunction expansion In region R 1 , the eigenfunction is represented as Y 1n ðyÞ ¼ cos½t n ðy þ aÞ�.Here, ϑ n denotes the wave number of the n-th mode and is defined as W n ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 À t 2 n p .The eigenvalues τ n satisfy the dispersion relation described by the following equation: In Eq (9), the first term represents the incident wave, while the second term represents the reflected field.The coefficients A n in the second term represent the unknown reflected mode coefficients.Additionally, the eigenfunctions Y 1n ðyÞ associated with this analysis also fulfill the orthogonality relation provided by: where, � m = 2 for m = 0 and 1 otherwise and δ mn is Kronecker delta.
0.2 In regions R 2 : fx < 0; À b < y < À ag and R 3 : fx < 0; a < y < bg The upper boundary of region R 2 is defined by the rigid wall condition at y = −a, as stated in Eq (4).On the other hand, the lower boundary at y = −b is governed by the impedance wall condition specified in Eq (5).Similarly, the lower boundary of region R 3 is determined by the rigid wall condition at y = a, as given in Eq (4).The upper boundary, on the other hand, is subject to the impedance wall condition at y = b, as described in Eq (6).By solving Eq (3) while considering these boundary conditions for regions R2 and R3, the eigenfunction expansion can be expressed in the following formulations: In this context, the eigenfunctions for regions R 2 and R 3 are given by Y 2n ðyÞ ¼ cos½l n ðy þ aÞ� and Y 3n ðyÞ ¼ cos½l n ðy À aÞ� respectively.The wave number associated with the n th mode can be mathematically represented as ϰ n ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi , where λ n denotes the n th eigenvalue.For the case of mixed boundary conditions at y = ±b, the eigenvalues for n = 0, 1, 2, � � � are determined as the roots of the following dispersion relation: In their respective domains, the eigenfunctions Y 2n ðyÞ and Y 3n ðyÞ are orthogonal to each other.This orthogonality is expressed through the following relations: where The region R 4 is bounded by impedance type conditions at y = ±h, as specified in Eq (6).Within this region, the eigenfunction expansion for the propagation of sound waves can be obtained by solving Eq (3) while considering the boundary conditions given in Eq (7).This allows for a comprehensive understanding of the behavior of sound waves within this region in term of following expansion Here Y 4n ðyÞ ¼ r sin ½g n ðy þ hÞ� þ sg n cos ½g n ðy þ hÞ� expresses the eigenfunction of a mode having mode wave number B n ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 À g 2 n p in which γ n are the eigenvalues.These eigenvalues are the roots of following dispersion relation: Moreover, the eigenfunctions Y 4n ðyÞ satisfy the orthogonality relation where Note that with impedance conditions, the roots of dispersion relations ( 14) and ( 18) must be found numerically and must be arranged in accordance with the properties as given in [14].Furthermore, the coefficients {A n , B n , C n , D n } are unknowns.To find these unknowns, we use matching interface conditions.

Matching interface conditions
Regarding the conditions governing interface matching, our focus is on ensuring the matching of pressure and normal velocity modes at the interface.To accurately capture the scattering response in the presence of impedance variations and geometric discontinuities, it is crucial to carefully consider the interface conditions.The literature provides several formulations of such conditions, as demonstrated by [14].Our specific attention is directed towards the aperture located at the interface, precisely at x = 0, where achieving consistency in pressure values across different regions is of paramount importance.Simultaneously, we integrate impedance discontinuities into the matching conditions for the velocity modes.We adopt the approach of maintaining the continuity of pressure modes, normalized with respect to the eigenfunctions of regions R 1 , R 2 , and R 3 , thereby providing a comprehensive framework for an accurate representation of the system.
By incorporating the eigenfunction expansions ( 9), ( 12), ( 13) and ( 17) into Eqs ( 21)-( 23), and subsequently solving the resultant equations with the assistance of the orthogonality relations outlined in ( 11) and ( 15), and following certain mathematical rearrangements, we derive the explicit expression for the scattering amplitudes as follows: • For region R 1 , we get where where • For region R 3 , we achieve where At the interface, we utilize the matching condition of normal velocities to determine the unidentified coefficient for region R4.Normalizing these conditions with respect to the eigenfunctions of region R4 results in: Utilizing the fluid potentials provided in ( 9), ( 12), ( 13) and (17) in Eq (30), and subsequently applying the orthogonality relation (19), one can obtain the explicit formulation of scattering amplitudes of region R4 through some mathematical rearrangements: where By applying Eqs ( 24), ( 26) and (28) into Eq (31), we obtain a linear algebraic system with unknowns D m , where m = 0, 1, 2, � � �.To determine these unknowns, the system is truncated and then inverted.Once the values for D m are determined, the quantities A n , B n , C n can be easily calculated using Eqs ( 24), ( 26) and (28).It is important to note that the system for the rigid discontinuities at x = 0 can be derived from Eq (31) by setting μ = 0.

Energy flux
Energy flux or power forms the basis for quantifying the distribution of energy across various aspects of the guiding structure, enabling a comprehensive understanding of its scattering behavior.The formulas for radiated energy flux, reflection, and transmission can be determined by applying the definition provided in [14], which is: where superscript asterisk (*) denotes the complex conjugate.By substituting the incident field e ix from Eq (9) into Eq (32), we can determine the incidence power as P inc ¼ a.Similarly, by substituting the reflecting and transmitting fields into Eq (32), we can calculate the power or energy flux in the duct sections Rj as follows: and Note that the negative sign in (33)-( 35) indicates that the powers are propagating in negative direction.The energy conservation law can be established by equating the powers propagating in negative and positive directions, that gives For analytical purposes, P inc is adjusted at unity, which is achieved on dividing (37) by a that is; where, E j ¼ À P j =a for j = 1, 2, 3 and E 4 ¼ P 4 =a.It's worth noting that Eq (38) is recognized as the conserved power identity, rooted in the principle of energy conservation.This identity implies that if one unit of power is input into the system, it will be equivalent to the combined sum of reflected and transmitted powers.

Numerical results and discussions
To solve the linear algebraic system presented in Eq (31), a numerical approach is employed by setting m = n = 0, 1, 2, � � �.This enables us to obtain the truncated amplitudes.In the numerical computations, a fluid density of ρ = 1.2043 kgm −3 and a sound speed of c = 343ms −1 are considered.Before delving into the scattering properties of the structure, it is important to evaluate the accuracy of the truncated solution.This can be achieved by numerically reconstructing the matching conditions and conservation law using the truncated form of the solution.To assess the accuracy, specific values are assigned to the parameters involved.In this case, we set , and N = 120, thereby establishing the structure setting and impedance.Furthermore, a frequency of f = 230Hz is chosen as the operating frequency.21)-( 23) and (30), indicating that the truncated amplitudes have adequately converged.Note that to analyze how changes in the geometric proportions of the duct structure affect the transmission and absorption of sound waves some specific parametric setting is used.The parameter settings, such as � b ¼ 3� a, were chosen to simulate realistic scenarios encountered in duct structures, where variations in dimensions are common.This parameter setting aligns with previous studies in the literature, for instance see [16], enabling comparisons and ensuring consistency in methodology.Furthermore, our aim is not only to model specific real-world scenarios but also to explore a range of plausible configurations to elucidate general trends and behaviors.Therefore, while the chosen parameters may not correspond directly to a particular practical problem, they serve the purpose of elucidating the physical behavior of the system under varied conditions, contributing to a deeper understanding of acoustic phenomena in duct structures., these abrupt variations diminish.This reduction is attributed to a decrease in the number of propagating modes supported by the system, leading to a smoother transition between transmission and reflection characteristics.The altered resonance spectrum, with cut-on modes now appearing at four different frequencies, underscores the sensitivity of the system to geometric parameters and highlights opportunities for optimizing its performance in practical applications.
To obtain the results depicted in   7, it can be seen that by changing the surface conditions form reflecting powers are changed.Therefore, one may see that by changing the surface conditions the scattering behavior can be optimized.Moreover, by using the step discontinuities the reflected powers are varied.The reason behind is the participation of more number of propagating modes with step-discontinuities involving  conditions leads to variations in reflecting powers, showcasing the potential for optimizing scattering behavior through such changes.Furthermore, the introduction of step discontinuities results in varying reflected powers.This phenomenon is attributed to the involvement of a greater number of propagating modes with step discontinuities compared to a planar structure.Specifically, for Figs 8(A) and 9(A) with step discontinuities, the number of cut-modes is 6 and 3, respectively, corresponding to values of � a around 0.387052, 0.782517, 0.875073, 1.41358, 1.91002, 2.03623m and 0.387052, 1.3042, 1.9605m.In contrast, with a planar setting (Figs 8(B) and 9(B)) and � h ¼ � b, the cut-on modes are 3 and 1, respectively.These cut-on modes are the primary contributors to abrupt variations in scattering powers.
The aforementioned findings provide significant insights into the intricate interplay between geometric parameters, surface conditions, and scattering characteristics within bifurcated structures.Our investigation, as depicted in Fig 5, reveals that varying the height of R 1 while maintaining fixed impedance parameters leads to a notable reduction in cut-on modes from seven to three as � b decreases from 3� a to 1:5� a.This reduction in cut-on modes correlates with diminished abrupt variations in scattering graphs, emphasizing the sensitivity of the system to geometric changes and suggesting avenues for optimization.Transitioning to Figs 6-9, where surface conditions are systematically altered, unveils the profound impact on scattering behavior.Distinct surface conditions yield diverse reflected powers, underscoring the potential for optimizing scattering behavior through strategic adjustments.Furthermore, the introduction of step discontinuities induces significant variations in reflected powers due to the involvement of a greater number of propagating modes.Specifically, the comparison between settings with and without step discontinuities highlights the pronounced influence on cut-on modes and subsequent abrupt variations in scattering powers.Overall, these findings offer valuable insights into optimizing the scattering behavior of bifurcated structures, holding promise for practical applications across engineering and physics domains.
The surfaces within the duct regions can be designed to absorb sound effectively by selecting specific parametric settings for impedance conditions.Relevant settings from existing literature, as indicated in [16], offer valuable insights into these parametric configurations.In this article, the chosen mixed parameters correspond to [16] with p = r = μ = 1, q = s = κ = iξ/k, where ξ defines the specific impedance as ξ = z + iη.Here, z and η represent the resistive and reactive components of the surface material.
To illustrate the impact of varying absorbent surfaces on transmission,

Concluding remarks
The current investigation delves into the wave scattering analysis of a planar trifurcated lined duct, considering diverse boundary properties, has yielded significant insights.The study extensively explored a spectrum of mixed boundary conditions, successfully addressing the governing problem.Through the integration of eigenfunctions, orthogonality relations, and matching conditions, the initially intricate differential system underwent a transformation into a numerically solvable linear algebraic system post-truncation.
The wave scattering behavior exhibited by the trifurcated lined duct under varying conditions, including alterations in boundary properties, duct size, and impedance discontinuity is studied.The results, derived in a generalized manner, effectively recaptured existing findings for the trifurcated lined duct as a distinct case.Specifically, we successfully recovered previously established results for varied boundary properties (soft, rigid, impedance) in scenarios devoid of structural discontinuity.Moreover, the analysis extended to the computation and scrutiny of radiated energy across all regions of the duct.A notable revelation emerged, indicating that lined ducts exhibited a reduced noise generation compared to their hard or soft counterparts.This finding underscores the practical advantage of employing lined duct configurations in situations where noise reduction is of paramount importance.
Additionally, the successful conservation of energy flux across diverse duct regions served as a validation of our algebraic approach.It confirmed the coherent propagation of cut-on duct modes within different sections, adding credibility to our methodology.Importantly, our study demonstrated that the simplified solution accurately recovered pressure and normal velocity modes, emphasizing the versatility and accuracy of our approach.

Figs 2 and 3
provide visual representations of the matching conditions for pressures and normal velocities at the interface x = 0 with respect to y. Fig 2 demonstrates that the real and imaginary parts of pressures, denoted as φ 4 (0, y), precisely align with φ 1 (0, y) within the domain |y| < a, φ 2 (0, y) within the domain −b < y < −a, and φ 3 (0, y) within the domain a < y < b.Similarly, Fig 3 illustrates that the real and imaginary components of the normal velocity, φ 4x (0, y), perfectly match at the aperture with φ 1x (0, y) in the domain |y| < a, φ 2x (0, y) in the domain −b < y < −a, and φ 3x (0, y) in the domain a < y < b.Additionally, along the impedance discontinuities, the real and imaginary components of the normal velocity, φ 4x (0, y), coincide with −φ 4 for κ = μ = 1 within the range −h < y < −b and b < y < h.This matching aligns with the assumptions made in Eqs (

Figs 4 and 5
illustrate the energy propagation in the waveguide with respect to frequency (f) and the variation in symmetric height discontinuities (a ¼ k � � a), respectively.To obtain the results depicted in Fig 4, we fix the height of region R 1 as � a ¼ 0:24m, while the impedance parameters are assumed to be r = s = p = q = 1.The dimensions of the other regions are defined

Fig 9 .
Fig 9. Reflected energies against a ¼ k � � a for rigid, soft and impedance conditions; (A) with step-discontinuities ( � h ¼ 5� a) (B) without-discontinuities ( � h ¼ � b), where � b ¼ 1:5� a. https://doi.org/10.1371/journal.pone.0306115.g009 Figs 10 and 11 have been generated.Results are presented against frequency and height for z = 0 and different values of η = −0.8,0, 1, 2. In Fig 10, transmitted powers are plotted against frequency with � b ¼ 3� a at � a ¼ 0:24m, while Fig 11 depicts transmitted powers against a ¼ k� a with � b ¼ 3� a at a frequency of f = 230Hz.The system is truncated with N = 50 terms.Notably, for Figs 10(A) and 11(A), � h ¼ 5� a was considered, while for Figs 10(B) and 11(B), � h ¼ � b was used.Cut-on modes occurred at (178, 214, 355, 715)Hz and (178, 355, 715)Hz for Fig 10(A) and 10(B), respectively.For Fig 11(A) and 11(B), cut-on modes occurred at � a � 0:782517m, 0.942386m, 1.56503m, and k � � a � 0:782517m, 1.56503m, respectively.It is evident that the presence of step-discontinuities leads to more propagating cut-on modes compared to results without discontinuities.Fig 11(B) closely aligns with the findings of Rawlins [16] using the Wiener-Hopf technique, supporting the mode-matching solution computed in the presence of step-discontinuities. Furthermore, the examination of Figs 10 and 11 provides crucial physical insights into the effect of varying absorbent surfaces on transmission characteristics within duct structures.By systematically manipulating the reactive component (η) while holding the resistive component constant (z = 0), these figures offer valuable insights into how different surface configurations influence the transmission of sound waves.Notably, Fig 10 presents transmitted powers plotted against frequency, offering a comprehensive view of how surface parameters impact transmission across a spectrum of frequencies, with � b ¼ 3� a and � a ¼ 0:24m.In contrast, Fig 11 illustrates transmitted powers against height (a ¼ k� a) at a fixed frequency of f = 230Hz, revealing the spatial variations in transmission resulting from diverse surface configurations.The identification of cut-on modes at specific frequencies in both figures further enriches our understanding.For Fig 10(A) and 10(B), where � h ¼ 5� a and � h ¼ � b respectively, cut-on modes manifest at distinct frequencies, highlighting how variations in surface conditions influence the resonant behavior of the system.Similarly, in Fig 11(A) and 11(B), where the height is varied with � b ¼ 3� a, cut-on modes occur at different heights, demonstrating the spatial dependence of resonance within the duct structure.These findings yield critical insights for designing and optimizing sound absorption systems, empowering engineers to tailor surface parameters to achieve desired transmission characteristics in practical applications.